While the linkup displays quickly, that delay usually happens when a script placed prior to the linkup script on the page loading sequence, takes quite a while to load.
In general, the linkup widget renders the links when the page is finished in order to have an estimate of the width of the page.
if some other script fails to load (e.g. the server that serves that other script is down or it is overly laggy), it makes the page finish later and thus the delay.
Dealing with that erroneous script makes both your site and the widget load faster.
One quick way to do so is to check the console of the browser (which can be brought up with F12 in most browsers). There is usually a "network" tab there
which, when the page loads, it graphs the loading times of each component.
Long gray or red lines are those erroneous scripts that take ages to load
on the pic of such a graph from a page, you can see one script taking a lot of time to be found and load, possibly causing some other script to lag behind.